Cullen is 10 years younger than Ada. The product of their ages 2 years ago was 39.

Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned by 10x2+4xâ1, 3xâ4x2+3, and

lord [1]

I suppose

$H=\mathrm{span}\{10x^2+4x-1,3x-4x^2+3,5x^2+x-1\}$

The vectors that span $H$ form a basis for $P_2$ if they are (1) linearly independent and (2) any vector in $P_2$ can be expressed as a linear combination of those vectors (i.e. they span $P_2$ ).

Independence:

Compute the Wronskian determinant:

$\begin{vmatrix}10x^2+4x-1&3x-4x^2+3&5x^2+x-1\\20x+4&3-8x&10x+1\\20&-8&10\end{vmatrix}=-6\neq0$

The determinant is non-zero, so the vectors are linearly independent. For this reason, we also know the dimension of $H$ is 3.

Span:

Write an arbitrary vector in $P_2$ as $ax^2+bx+c$ . Then the given vectors span $P_2$ if there is always a choice of scalars $k_1,k_2,k_3$ such that

$k_1(10x^2+4x-1)+k_2(3x-4x^2+3)+k_3(5x^2+x-1)=ax^2+bx+c$

which is equivalent to the system

$\begin{bmatrix}10&-4&5\\4&3&1\\-1&3&-1\end{bmatrix}\begin{bmatrix}k_1\\k_2\\k_3\end{bmatrix}=\begin{bmatrix}a\\b\\c\end{bmatrix}$

The coefficient matrix is non-singular, so it has an inverse. Multiplying both sides by that inverse gives

$\begin{bmatrix}k_1\\k_2\\k_3\end{bmatrix}=\begin{bmatrix}-\dfrac{6a-11b+19c}3\\\dfrac{3a-5b+2c}3\\\dfrac{15a-26b+46c}3\end{bmatrix}$

so the vectors do span $P_2$ .

The vectors comprising $H$ form a basis for it because they are linearly independent.

4 0

2 years ago

Which expressions are equivalent to the given expression √40

german

Answer:

2√10

Step-by-step explanation:

√40 = √(4*10) = √4 * √10 = 2√10

6 0

1 year ago

Read 2 more answers

N(17+x)=34x−r<br> I need to solve for x

Paraphin [41]

Answer:

The value of x is, $x= \frac{17N+r}{34-N}$

Explanation:

Given: $N(17+x)=34x-r$

Distributive Property states that when a number is multiplied by the sum of two numbers, the first number can be distributed to both of those numbers and multiplied by each of them separately.

If $a\cdot(b+c) =a\cdot b + a\cdot c$

Now, using distributive property on left hand side of the given expression as:

$N\cdot 17+N\cdot x = 34x-r$ or $17N+Nx = 34x-r$

Addition Property of equality state that we add the same number from both sides of an equation.

Add r to both sides of an equation:

$17N+Nx+r=34x-r+r$

Simplify:

$17N+Nx+r=34x$

Subtraction Property of equality state that we subtract the same number from both sides of an equation.

Subtract Nx from both sides of an equation;

$17N+Nx+r-Nx=34x-Nx$

Simplify:

$17N+r=34x-Nx$

or

$17N+r=x(34-N)$

Division Property of equality states that we divide the same number from both sides of an equation.

Divide by (34-N) to both sides of an equation;

$\frac{17N+r}{34-N}= \frac{x(34-N)}{34-N}$

On Simplify:

$x= \frac{17N+r}{34-N}$

4 0

2 years ago

Read 2 more answers

A researcher in environmental science is conducting a study to investigate the impact of a particular herbicide on fish. He has

Vedmedyk [2.9K]

Answer:

The herbicide has no impact on fish

Step-by-step explanation:

Experimental research includes to divide the concerned group 'fishes' here into 'treatment' & 'control' group, to study impact of a treatment. Hypothesis test the statistical significance of propositions. Null hypothesis signifies no impact, alternate hypothesis signifies impact.

Let averages enzyme level of treatment & control be Xt & Xc

Null Hypothesis [H0] : Xt = Xc, or Xt - Xc = 0
Alternate Hypothesis [H1] : Xt ≠ Xc, or Xt - Xc ≠ 0

If hypothesis test suggests that there is no significant difference between treatment & control group enzyme level, so we reject H0 & accept H1. It implies that the herbicide has no significant impact on fishes enzymes.

8 0

2 years ago

A new car is purchased for 15300 dollars. The value of the car depreciates at 14.25% per year. What will the value of the car be

Andrei [34K]

Answer:

$\$6,082.70$